Average Error: 0.0 → 0.0
Time: 12.2s
Precision: binary64
Cost: 19776
\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
↓
\[\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right) \]
(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))
↓
(FPCore (x y z t a b c i)
 :precision binary64
 (fma x y (fma z t (fma c i (* a b)))))
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((x * y) + (z * t)) + (a * b)) + (c * i);
}
↓
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(x, y, fma(z, t, fma(c, i, (a * b))));
}

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(x * y) + Float64(z * t)) + Float64(a * b)) + Float64(c * i))
end
↓
function code(x, y, z, t, a, b, c, i)
	return fma(x, y, fma(z, t, fma(c, i, Float64(a * b))))
end

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(x * y + N[(z * t + N[(c * i + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i
↓
\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)

Error

Derivation

  1. Initial program 0.0

    \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)} \]
    Proof
    (fma.f64 x y (fma.f64 z t (fma.f64 c i (*.f64 a b)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 x y (fma.f64 z t (Rewrite<= fma-def_binary64 (+.f64 (*.f64 c i) (*.f64 a b))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 x y (fma.f64 z t (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 a b) (*.f64 c i))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 x y (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z t) (+.f64 (*.f64 a b) (*.f64 c i))))): 2 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) (+.f64 (*.f64 z t) (+.f64 (*.f64 a b) (*.f64 c i))))): 2 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (+.f64 (*.f64 a b) (*.f64 c i)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) (*.f64 c i))): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right) \]

Alternatives

Alternative 1
Error22.0
Cost2788
\[\begin{array}{l} t_1 := c \cdot i + x \cdot y\\ t_2 := a \cdot b + x \cdot y\\ t_3 := a \cdot b + z \cdot t\\ t_4 := x \cdot y + z \cdot t\\ \mathbf{if}\;c \cdot i \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq -3.5 \cdot 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq -8.5 \cdot 10^{-204}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \cdot i \leq -1.02 \cdot 10^{-284}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 3.4 \cdot 10^{-301}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \cdot i \leq 2.5 \cdot 10^{-168}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 2.6 \cdot 10^{-36}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \cdot i \leq 62000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 9.5 \cdot 10^{+21}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error38.2
Cost2272
\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq -2.4 \cdot 10^{-62}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq -2.2 \cdot 10^{-211}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq -1.18 \cdot 10^{-284}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 2.3 \cdot 10^{-248}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 1.62 \cdot 10^{-173}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 1.8 \cdot 10^{-85}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 1.22 \cdot 10^{-17}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
Alternative 3
Error22.0
Cost1748
\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ t_2 := a \cdot b + z \cdot t\\ t_3 := a \cdot b + c \cdot i\\ \mathbf{if}\;c \cdot i \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \cdot i \leq 3.5 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 1.85 \cdot 10^{-42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 0.9:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 8 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 4
Error21.9
Cost1748
\[\begin{array}{l} t_1 := a \cdot b + x \cdot y\\ t_2 := a \cdot b + z \cdot t\\ \mathbf{if}\;c \cdot i \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 1.65 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 6 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 0.0255:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 4.6 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \end{array} \]
Alternative 5
Error22.1
Cost1748
\[\begin{array}{l} t_1 := c \cdot i + x \cdot y\\ t_2 := a \cdot b + z \cdot t\\ \mathbf{if}\;c \cdot i \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 3.5 \cdot 10^{-162}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 1.8 \cdot 10^{-36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \cdot i \leq 64000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 1.75 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error32.9
Cost1504
\[\begin{array}{l} t_1 := a \cdot b + c \cdot i\\ t_2 := a \cdot b + z \cdot t\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{-117}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+194}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+231}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
Alternative 7
Error16.2
Cost1364
\[\begin{array}{l} t_1 := a \cdot b + \left(c \cdot i + x \cdot y\right)\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{-14}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+79}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+244}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error37.8
Cost1232
\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 9 \cdot 10^{-162}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;c \cdot i \leq 8.5 \cdot 10^{-83}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;c \cdot i \leq 1.3 \cdot 10^{-17}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
Alternative 9
Error6.0
Cost1224
\[\begin{array}{l} t_1 := a \cdot b + \left(c \cdot i + x \cdot y\right)\\ \mathbf{if}\;c \cdot i \leq -1.05 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \cdot i \leq 1.8 \cdot 10^{+22}:\\ \;\;\;\;a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error32.8
Cost976
\[\begin{array}{l} t_1 := a \cdot b + c \cdot i\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{-113}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+195}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+232}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
Alternative 11
Error0.0
Cost960
\[a \cdot b + \left(c \cdot i + \left(x \cdot y + z \cdot t\right)\right) \]
Alternative 12
Error38.2
Cost712
\[\begin{array}{l} \mathbf{if}\;c \cdot i \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 2.7 \cdot 10^{-17}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \]
Alternative 13
Error47.1
Cost192
\[a \cdot b \]

Error

Reproduce

herbie shell --seed 2022329 
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3, C"
  :precision binary64
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))